Indexed type families, or '''type families''' for short, are a Haskell extension supporting ad-hoc overloading of data types. Type families are parametric types that can be assigned specialized representations based on the type parameters they are instantiated with. They are the data type analogue of [[Type class|type classes]]: families are used to define overloaded ''data'' in the same way that classes are used to define overloaded ''functions''. Type families are useful for generic programming, for creating highly parameterised library interfaces, and for creating interfaces with enhanced static information, much like dependent types.

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Type families come in two flavors: ''data families'' and ''type synonym families''. Data families are the indexed form of data and newtype definitions. Type synonym families are the indexed form of type synonyms. Each of these flavors can be defined in a standalone manner or ''associated'' with a type class. Standalone definitions are more general, while associated types can more clearly express how a type is used and lead to better error messages.

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''NB: see also Simon's [http://hackage.haskell.org/trac/ghc/blog/LetGeneralisationInGhc7 blog entry on let generalisation] for a significant change in the policy for let generalisation, driven by the type family extension. In brief: a few programs will puzzlingly fail to compile with <tt>-XTypeFamilies</tt> even though the code is legal Haskell 98.''

== What are type families? ==

== What are type families? ==

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Indexed type families, or type families for short, are type constructors that represent ''sets of types.'' Set members are denoted by supplying the type family constructor with type parameters, which are called ''type indices''. The difference between vanilla parametrised type constructors and family constructors is much like between parametrically polymorphic functions and (ad-hoc polymorphic) methods of type classes. Parametric polymorphic functions behave the same at all type instances, whereas class methods can change their behaviour in dependence on the class type parameters. Similarly, vanilla type constructors imply the same data representation for all type instances, but family constructors can have varying representation types for varying type indices.

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The concept of a type family comes from type theory. An indexed type family in type theory is a partial function at the type level. Applying the function to parameters (called ''type indices'') yields a type. Type families permit a program to compute what data constructors it will operate on, rather than having them fixed statically (as with simple type systems) or treated as opaque unknowns (as with parametrically polymorphic types).

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Indexed type families come in two flavours: ''data families'' and ''type synonym families''. They are the indexed family variants of algebraic data types and type synonyms, respectively. The instances of data families can be data types and newtypes.

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Type families are to vanilla data types what type class methods are to regular functions. Vanilla polymorphic data types and functions have a single definition, which is used at all type instances. Classes and type families, on the other hand, have an interface definition and any number of instance definitions. A type family's interface definition declares its [[kind]] and its arity, or the number of type indices it takes. Instance definitions define the type family over some part of the domain.

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As a simple example of how type families differ from ordinary parametric data types, consider a strict list type. We can represent a list of <hask>Char</hask> in the usual way, with cons cells. We can do the same thing to represent a list of <hask>()</hask>, but since a strict <hask>()</hask> value carries no useful information, it would be more efficient to just store the length of the list. This can't be done with an ordinary parametric data type, because the data constructors used to represent the list would depend on the list's type parameter: if it's <hask>Char</hask> then the list consists of cons cells; if it's <hask>()</hask>, then the list consists of a single integer. We basically want to select between two different data types based on a type parameter. Using type families, this list type could be declared as follows:

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<haskell>

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-- Declare a list-like data family

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data family XList a

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-- Declare a list-like instance for Char

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data instance XList Char = XCons !Char !(XList Char) | XNil

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-- Declare a number-like instance for ()

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data instance XList () = XListUnit !Int

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</haskell>

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== Type families in GHC ==

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The right-hand sides of the two <code>data instance</code> declarations are exactly ordinary data definitions. In fact, a <code>data instance</code> declaration is nothing more than a shorthand for a <code>data</code> declaration followed by a <code>type instance</code> (see below) declaration. However, they define two instances of the same parametric data type, <hask>XList Char</hask> and <hask>XList ()</hask>, whereas ordinary data declarations define completely unrelated types. A recent [[Simonpj/Talk:FunWithTypeFuns|tutorial paper]] has more in-depth examples of programming with type families.

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Indexed type families are a new GHC extension to facilitate type-level programming. Type families are a generalisation of [http://www.cse.unsw.edu.au/~chak/papers/CKPM05.html associated data types] and [http://www.cse.unsw.edu.au/~chak/papers/CKP05.html associated type synonyms], and are described in a [http://www.cse.unsw.edu.au/~chak/papers/SPCS08.html recent ICFP paper]. They essentially provide type-indexed data types and named functions on types, which are useful for generic programming and highly parameterised library interfaces as well as interfaces with enhanced static information, much like dependent types. They might also be regarded as an alternative to functional dependencies, but provide a more functional style of type-level programming than the relational style of functional dependencies.

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[[GADT]]s bear some similarity to type families, in the sense that they allow a parametric type's constructors to depend on the type's parameters. However, all GADT constructors must be defined in one place, whereas type families can be extended. Functional dependences are similar to type families, and many type classes that use functional dependences can be equivalently expressed with type families. Type families provide a more functional style of type-level programming than the relational style of functional dependences.

== What do I need to use type families? ==

== What do I need to use type families? ==

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The first stable release of GHC that properly supports type families is 6.10.1. (An early partial implementation was part of the 6.8 release, but its use is deprecated.) Please [http://hackage.haskell.org/trac/ghc/query?status=new&status=assigned&status=reopened&group=priority&type=bug&order=id&desc=1 report bugs] via the GHC bug tracker, ideally accompanied by a small example program that demonstrates the problem. Use the [mailto:glasgow-haskell-users@haskell.org GHC mailing list] for questions or for a discussion of this language extension and its description on this wiki page.

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Type families are a GHC extension enabled with the <code>-XTypeFamilies</code> flag or the <code>{-# LANGUAGE TypeFamilies #-}</code> pragma. The first stable release of GHC that properly supports type families is 6.10.1. (The 6.8 release included an early partial implementation, but its use is deprecated.) Please [http://hackage.haskell.org/trac/ghc/query?status=new&status=assigned&status=reopened&group=priority&type=bug&order=id&desc=1 report bugs] via the GHC bug tracker, ideally accompanied by a small example program that demonstrates the problem. Use the [mailto:glasgow-haskell-users@haskell.org GHC mailing list] for questions or for a discussion of this language extension and its description on this wiki page.

== An associated data type example ==

== An associated data type example ==

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As an example, consider Ralf Hinze's [http://www.informatik.uni-bonn.de/~ralf/publications.html#J4 generalised tries], a form of generic finite maps.

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As an example, consider Ralf Hinze's [http://www.cs.ox.ac.uk/ralf.hinze/publications/GGTries.ps.gz generalised tries], a form of generic finite maps.

=== The class declaration ===

=== The class declaration ===

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insert :: k -> v -> GMap k v -> GMap k v

insert :: k -> v -> GMap k v -> GMap k v

</haskell>

</haskell>

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The interesting part is the ''associated data family'' declaration of the class. It gives a [http://www.haskell.org/ghc/docs/latest/html/users_guide/type-families.html#data-family-declarations ''kind signature''] (here <hask>* -> *</hask>) for the associated data type <hask>GMap k</hask> - analog to how methods receive a type signature in a class declaration.

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The interesting part is the ''associated data family'' declaration of the class. It gives a [http://www.haskell.org/ghc/docs/latest/html/users_guide/type-families.html#data-family-declarations ''kind signature''] (here <hask>* -> *</hask>) for the associated data type <hask>GMap k</hask> - analogous to how methods receive a type signature in a class declaration.

What it is important to notice is that the first parameter of the associated type <hask>GMap</hask> coincides with the class parameter of <hask>GMapKey</hask>. This indicates that also in all instances of the class, the instances of the associated data type need to have their first argument match up with the instance type. In general, the type arguments of an associated type can be a subset of the class parameters (in a multi-parameter type class) and they can appear in any order, possibly in an order other than in the class head. The latter can be useful if the associated data type is partially applied in some contexts.

What it is important to notice is that the first parameter of the associated type <hask>GMap</hask> coincides with the class parameter of <hask>GMapKey</hask>. This indicates that also in all instances of the class, the instances of the associated data type need to have their first argument match up with the instance type. In general, the type arguments of an associated type can be a subset of the class parameters (in a multi-parameter type class) and they can appear in any order, possibly in an order other than in the class head. The latter can be useful if the associated data type is partially applied in some contexts.

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=== An Int instance ===

=== An Int instance ===

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To use Ints as keys into generic maps, we declare an instance that simply uses <hask>Data.Map</hask>, thusly:

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To use Ints as keys into generic maps, we declare an instance that simply uses <hask>Data.IntMap</hask>, thusly:

If you find this code algorithmically surprising, I'd suggest to have a look at [http://www.informatik.uni-bonn.de/~ralf/publications.html#J4 Ralf Hinze's paper]. The only novelty concerning associated types, in these two instances, is that the instances have a context <hask>(GMapKey a, GMapKey b)</hask>. Consequently, the right hand sides of the associated type declarations can use <hask>GMap</hask> recursively at the key types <hask>a</hask> and <hask>b</hask> - not unlike the method definitions use the class methods recursively at the types for which the class is given in the instance context.

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If you find this code algorithmically surprising, I'd suggest to have a look at [http://www.cs.ox.ac.uk/ralf.hinze/publications/index.html#J4 Ralf Hinze's paper]. The only novelty concerning associated types, in these two instances, is that the instances have a context <hask>(GMapKey a, GMapKey b)</hask>. Consequently, the right hand sides of the associated type declarations can use <hask>GMap</hask> recursively at the key types <hask>a</hask> and <hask>b</hask> - not unlike the method definitions use the class methods recursively at the types for which the class is given in the instance context.

=== Using a generic map ===

=== Using a generic map ===

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=== Download the code ===

=== Download the code ===

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If you want to play with this example without copying it off the wiki, just download the [http://darcs.haskell.org/testsuite/tests/ghc-regress/indexed-types/should_run/GMapAssoc.hs source code for <hask>GMap</hask>] from GHC's test suite.

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If you want to play with this example without copying it off the wiki, just download the source code[http://darcs.haskell.org/testsuite/tests/ghc-regress/indexed-types/should_run/GMapAssoc.hs] for <hask>GMap</hask> from GHC's test suite.

== Detailed definition of data families ==

== Detailed definition of data families ==

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<haskell>

<haskell>

instance (GMapKey a, GMapKey b) => GMapKey (Either a b) where

instance (GMapKey a, GMapKey b) => GMapKey (Either a b) where

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data GMap (Either a b) v = GMapEither (GMap a v) (GMap b v

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data GMap (Either a b) v = GMapEither (GMap a v) (GMap b v)

...

...

</haskell>

</haskell>

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F Bool -- WRONG: unsaturated application

F Bool -- WRONG: unsaturated application

</haskell>

</haskell>

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A top-level type family can be declared as open or closed. (Associated type

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families are always open.) A closed type family has all of its equations

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defined in one place and cannot be extended, whereas an open family can have

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instances spread across modules. The advantage of a closed family is that

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its equations are tried in order, similar to a term-level function definition:

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<haskell>

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type family G a where

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G Int = Bool

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G a = Char

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</haskell>

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With this definition, the type <hask>G Int</hask> becomes <hask>Bool</hask>

Instance declarations of type families are very similar to standard type synonym declarations. The only two differences are that the keyword <hask>type</hask> is followed by <hask>instance</hask> and that some or all of the type arguments can be non-variable types, but may not contain forall types or type synonym families. However, data families are generally allowed, and type synonyms are allowed as long as they are fully applied and expand to a type that is admissible - these are the exact same requirements as for data instances. For example, the <hask>[e]</hask> instance for <hask>Elem</hask> is

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Instance declarations of open type families are very similar to standard type synonym declarations. The only two differences are that the keyword <hask>type</hask> is followed by <hask>instance</hask> and that some or all of the type arguments can be non-variable types, but may not contain forall types or type synonym families. However, data families are generally allowed, and type synonyms are allowed as long as they are fully applied and expand to a type that is admissible - these are the exact same requirements as for data instances. For example, the <hask>[e]</hask> instance for <hask>Elem</hask> is

<haskell>

<haskell>

type instance Elem [e] = e

type instance Elem [e] = e

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* The right-hand side of a type instance must be a monotype (i.e., it may not include foralls) and after the expansion of all saturated vanilla type synonyms, no synonyms, except family synonyms may remain.

* The right-hand side of a type instance must be a monotype (i.e., it may not include foralls) and after the expansion of all saturated vanilla type synonyms, no synonyms, except family synonyms may remain.

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Here are some examples of admissible and illegal type instances:

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Here are some examples of admissible and illegal type instances and closed families:

Currently in recent versions of ghc 7.7 and planed to be included in 7.8.1.

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When dealing with closed families, simplifying the type is harder than just finding a left-hand side that matches and replacing that with a right-hand side. GHC will select an equation to use in a given type family application (the "target") if and only if the following 2 conditions hold:

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# There is a substitution from the variables in the equation's LHS that makes the left-hand side of the branch coincide with the target.

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# For each previous equation in the family: either the LHS of that equation is ''apart'' from the type family application, '''or''' the equation is ''compatible'' with the chosen equation.

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Now, we define ''apart'' and ''compatible'':

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# Two types are ''apart'' when one cannot simplify to the other, even after arbitrary type-family simplifications

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# Two equations are ''compatible'' if, either, their LHSs are apart or their LHSs unify and their RHSs are the same under the substitution induced by the unification.

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Some examples are in order:

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<haskell>

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type family F a where

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F Int = Bool

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F Bool = Char

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F a = Bool

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type family And (a :: Bool) (b :: Bool) :: Bool where

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And False c = False

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And True d = d

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And e False = False

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And f True = f

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And g g = g

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</haskell>

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In <hask>F</hask>, all pairs of equations are compatible except the second and third. The first two are compatible because their LHSs are apart. The first and third are compatible because the unifying substitution leads the RHSs to be the same. But, the second and third are not compatible because neither of these conditions holds. As a result, GHC will not use the third equation to simplify a target unless that target is apart from <hask>Bool</hask>.

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In <hask>And</hask>, ''every'' pair of equations is compatible, meaning GHC never has to make the extra apartness check during simplification.

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Why do all of this? It's a matter of type safety. Consider this example:

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<haskell>

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type family J a b

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type instance where

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J a a = Int

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J a b = Bool

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</haskell>

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Say GHC selected the second branch just because the first doesn't apply at the moment, because two type variables are distinct. The problem is that those variables might later be instantiated at the same value, and then the first branch would have applied. You can convince this sort of inconsistency to produce <hask>unsafeCoerce</hask>.

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It gets worse. GHC has no internal notion of inequality, so it can't use previous, failed term-level GADT pattern matches to refine its type assumptions. For example:

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<haskell>

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data G :: * -> * where

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GInt :: G Int

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GBool :: G Bool

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type family Foo (a :: *) :: *

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type instance where

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Foo Int = Char

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Foo a = Double

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bar :: G a -> Foo a

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bar GInt = 'x'

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bar _ = 3.14

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</haskell>

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The last line will fail to typecheck, because GHC doesn't know that the type variable <hask>a</hask> can't be <hask>Int</hask> here, even though it's obvious. The only general way to fix this is to have inequality evidence introduced into GHC, and that's a big deal and we don't know if we have the motivation for such a change yet.

==== Associated type instances ====

==== Associated type instances ====

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==== Overlap ====

==== Overlap ====

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The instance declarations of a type family used in a single program may only overlap if the right-hand sides of the overlapping instances coincide for the overlapping types. More formally, two instance declarations overlap if there is a substitution that makes the left-hand sides of the instances syntactically the same. Whenever that is the case, the right-hand sides of the instances must also be syntactically equal under the same substitution. This condition is independent of whether the type family is associated or not, and it is not only a matter of consistency, but one of type safety.

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The instance declarations of an open type family used in a single program must be compatible, in the form defined above. This condition is independent of whether the type family is associated or not, and it is not only a matter of consistency, but one of type safety.

Here are two examples to illustrate the condition under which overlap is permitted.

Here are two examples to illustrate the condition under which overlap is permitted.

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</haskell>

</haskell>

That is, we represent every functional dependency (FD) <hask>a1 .. an -> b</hask> by an FD type family <hask>F a1 .. an</hask> and a superclass context equality <hask>F a1 .. an ~ b</hask>, essentially giving a name to the functional dependency. In class instances, we define the type instances of FD families in accordance with the class head. Method signatures are not affected by that process.

That is, we represent every functional dependency (FD) <hask>a1 .. an -> b</hask> by an FD type family <hask>F a1 .. an</hask> and a superclass context equality <hask>F a1 .. an ~ b</hask>, essentially giving a name to the functional dependency. In class instances, we define the type instances of FD families in accordance with the class head. Method signatures are not affected by that process.

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NB: Equalities in superclass contexts are not fully implemented in the GHC 6.10 branch.

== Frequently asked questions ==

== Frequently asked questions ==

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=== Comparing type families and functional dependencies ===

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Functional dependencies cover some of the same territory as type families. How do the two compare?

* [http://hackage.haskell.org/trac/ghc/wiki/TFvsFD GHC trac] on a comparison of functional dependencies and type families

=== Injectivity, type inference, and ambiguity ===

=== Injectivity, type inference, and ambiguity ===

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Couldn't match expected type `F Int' against inferred type `F a1'

Couldn't match expected type `F Int' against inferred type `F a1'

</haskell>

</haskell>

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In type-checking <tt>g</tt>'s right hand side GHC discovers (by instantiating f's type with a fresh type variable) that it has type <tt>F a1 -> F a1</tt> for some as-yet-unknown type <tt>a1</tt>. Now it tries to make the inferred type match <tt>g</tt>'s type signature. Well, you say, just make <haskell>a1</haskell> equal to <tt>Int</tt> and you are done. True, but what if there were these instance

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In type-checking <tt>g</tt>'s right hand side GHC discovers (by instantiating <tt>f</tt>'s type with a fresh type variable) that it has type <tt>F a1 -> F a1</tt> for some as-yet-unknown type <tt>a1</tt>. Now it tries to make the inferred type match <tt>g</tt>'s type signature. Well, you say, just make <tt>a1</tt> equal to <tt>Int</tt> and you are done. True, but what if there were these instances

<haskell>

<haskell>

type instance F Int = Bool

type instance F Int = Bool

type instance F Char = Bool

type instance F Char = Bool

</haskell>

</haskell>

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Then making <tt>a1</tt> equal to <tt>Char</tt> would ''also'' make the two types equal. Because there is more than one choice, the program is rejected.

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Then making <tt>a1</tt> equal to <tt>Char</tt> would ''also'' make the two types equal. Because there is (potentially) more than one choice, the program is rejected.

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However (and confusingly) if you omit the type signature on <tt>g</tt> altogether, thus

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<haskell>

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f :: F a -> F a

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f = undefined

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g x = f x

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</haskell>

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GHC will happily infer the type <tt>g :: F a -> F a</tt>. But you can't ''write'' that type signature or, indeed, the more specific one above. (Arguably this behaviour, where GHC ''infers'' a type it can't ''check'', is very confusing. I suppose we could make GHC reject both programs, with and without type signatures.)

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'''What is the problem?''' The nub of the issue is this: knowing that <tt>F t1</tt>=<tt>F t2</tt> does ''not'' imply that <tt>t1</tt> = <tt>t2</tt>.

The difficulty is that the type function <tt>F</tt> need not be ''injective''; it can map two distinct types to the same type. For an injective type constructor like <tt>Maybe</tt>, if we know that <tt>Maybe t1</tt> = <tt>Maybe t2</tt>, then we know that <tt>t1</tt> = <tt>t2</tt>. But not so for non-injective type functions.

The difficulty is that the type function <tt>F</tt> need not be ''injective''; it can map two distinct types to the same type. For an injective type constructor like <tt>Maybe</tt>, if we know that <tt>Maybe t1</tt> = <tt>Maybe t2</tt>, then we know that <tt>t1</tt> = <tt>t2</tt>. But not so for non-injective type functions.

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The only solution is to avoid ambiguous types. In the type signature of a function,

The only solution is to avoid ambiguous types. In the type signature of a function,

* Ensure that every type variable occurs in the part after the "<tt>=></tt>"

* Ensure that every type variable occurs in the part after the "<tt>=></tt>"

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* Ensure that every type variable appears at least once outside a type function call.

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* Ensure that every type variable appears at least once outside a type function call.

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Alternatively, you can use data families, which create new types and are therefore injective. The following code works:

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Even then ambiguity is possible. For example:

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<haskell>data family F a

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<haskell>

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f :: F a -> [a]

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f :: F a -> F a

f = undefined

f = undefined

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−

g :: F b -> Int

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g :: F Int -> F Int

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g x = length (f x)

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g x = f x</haskell>

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</haskell>

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Although <tt>f</tt>'s type is unambiguous, its result type is swallowed up by <tt>length</tt>, which now makes <tt>g</tt>'s type ambiguous.

* [[Simonpj/Talk:FunWithTypeFuns | Fun with Type Functions]] Oleg Kiselyov, Simon Peyton Jones, Chung-chieh Shan (the source for this paper can be found at http://patch-tag.com/r/schoenfinkel/typefunctions/wiki )

[[Category:Type-level programming]]

[[Category:Type-level programming]]

−

[[Category:Language extension]]

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[[Category:Language extensions]]

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[[Category:GHC|Indexed types]]

Revision as of 08:56, 14 October 2013

Indexed type families, or type families for short, are a Haskell extension supporting ad-hoc overloading of data types. Type families are parametric types that can be assigned specialized representations based on the type parameters they are instantiated with. They are the data type analogue of type classes: families are used to define overloaded data in the same way that classes are used to define overloaded functions. Type families are useful for generic programming, for creating highly parameterised library interfaces, and for creating interfaces with enhanced static information, much like dependent types.

Type families come in two flavors: data families and type synonym families. Data families are the indexed form of data and newtype definitions. Type synonym families are the indexed form of type synonyms. Each of these flavors can be defined in a standalone manner or associated with a type class. Standalone definitions are more general, while associated types can more clearly express how a type is used and lead to better error messages.

NB: see also Simon's blog entry on let generalisation for a significant change in the policy for let generalisation, driven by the type family extension. In brief: a few programs will puzzlingly fail to compile with -XTypeFamilies even though the code is legal Haskell 98.

1 What are type families?

The concept of a type family comes from type theory. An indexed type family in type theory is a partial function at the type level. Applying the function to parameters (called type indices) yields a type. Type families permit a program to compute what data constructors it will operate on, rather than having them fixed statically (as with simple type systems) or treated as opaque unknowns (as with parametrically polymorphic types).

Type families are to vanilla data types what type class methods are to regular functions. Vanilla polymorphic data types and functions have a single definition, which is used at all type instances. Classes and type families, on the other hand, have an interface definition and any number of instance definitions. A type family's interface definition declares its kind and its arity, or the number of type indices it takes. Instance definitions define the type family over some part of the domain.

As a simple example of how type families differ from ordinary parametric data types, consider a strict list type. We can represent a list of

Char

in the usual way, with cons cells. We can do the same thing to represent a list of

()

, but since a strict

()

value carries no useful information, it would be more efficient to just store the length of the list. This can't be done with an ordinary parametric data type, because the data constructors used to represent the list would depend on the list's type parameter: if it's

Char

then the list consists of cons cells; if it's

()

, then the list consists of a single integer. We basically want to select between two different data types based on a type parameter. Using type families, this list type could be declared as follows:

The right-hand sides of the two data instance declarations are exactly ordinary data definitions. In fact, a data instance declaration is nothing more than a shorthand for a data declaration followed by a type instance (see below) declaration. However, they define two instances of the same parametric data type,

GADTs bear some similarity to type families, in the sense that they allow a parametric type's constructors to depend on the type's parameters. However, all GADT constructors must be defined in one place, whereas type families can be extended. Functional dependences are similar to type families, and many type classes that use functional dependences can be equivalently expressed with type families. Type families provide a more functional style of type-level programming than the relational style of functional dependences.

2 What do I need to use type families?

Type families are a GHC extension enabled with the -XTypeFamilies flag or the {-# LANGUAGE TypeFamilies #-} pragma. The first stable release of GHC that properly supports type families is 6.10.1. (The 6.8 release included an early partial implementation, but its use is deprecated.) Please report bugs via the GHC bug tracker, ideally accompanied by a small example program that demonstrates the problem. Use the GHC mailing list for questions or for a discussion of this language extension and its description on this wiki page.

3 An associated data type example

As an example, consider Ralf Hinze's generalised tries, a form of generic finite maps.

3.1 The class declaration

We define a type class whose instances are the types that we can use as keys in our generic maps:

The interesting part is the associated data family declaration of the class. It gives a kind signature (here

*->*

) for the associated data type

GMap k

- analogous to how methods receive a type signature in a class declaration.
What it is important to notice is that the first parameter of the associated type

GMap

coincides with the class parameter of

GMapKey

. This indicates that also in all instances of the class, the instances of the associated data type need to have their first argument match up with the instance type. In general, the type arguments of an associated type can be a subset of the class parameters (in a multi-parameter type class) and they can appear in any order, possibly in an order other than in the class head. The latter can be useful if the associated data type is partially applied in some contexts.
The second important point is that as

GMap k

has kind

*->*

and

k

(implicitly) has kind

*

, the type constructor

GMap

(without an argument) has kind

*->*->*

. Consequently, we see that

GMap

is applied to two arguments in the signatures of the methods

empty

,

lookup

, and

insert

.

3.2 An Int instance

To use Ints as keys into generic maps, we declare an instance that simply uses

needs to have both of its parameters, but as only the first one corresponds to a class parameter, the second needs to be a type variable (here

v

). As mentioned before, any associated type parameter that corresponds to a class parameter must be identical to the class parameter in each instance. The right hand side of the associated data declaration is like that of any other data type.

NB: At the moment, GADT syntax is not allowed for associated data types (or other indexed types). This is not a fundamental limitation, but just a shortcoming of the current implementation, which we expect to lift in the future.

As an exercise, implement an instance for

Char

that maps back to the

Int

instance using the conversion functions

Char.ord

and

Char.chr

.

3.3 A unit instance

Generic definitions, apart from elementary types, typically cover units, products, and sums. We start here with the unit instance for

If you find this code algorithmically surprising, I'd suggest to have a look at Ralf Hinze's paper. The only novelty concerning associated types, in these two instances, is that the instances have a context

(GMapKey a, GMapKey b)

. Consequently, the right hand sides of the associated type declarations can use

GMap

recursively at the key types

a

and

b

- not unlike the method definitions use the class methods recursively at the types for which the class is given in the instance context.

3.5 Using a generic map

Finally, some code building and querying a generic map:

myGMap :: GMap (Int,EitherChar())String
myGMap = insert (5, Left 'c')"(5, Left 'c')"$
insert (4, Right ())"(4, Right ())"$
insert (5, Right ())"This is the one!"$
insert (5, Right ())"This is the two!"$
insert (6, Right ())"(6, Right ())"$
insert (5, Left 'a')"(5, Left 'a')"$
empty
main =putStrLn$maybe"Couldn't find key!"id$lookup(5, Right ()) myGMap

3.6 Download the code

If you want to play with this example without copying it off the wiki, just download the source code[1] for

GMap

from GHC's test suite.

4 Detailed definition of data families

Data families appear in two flavours: (1) they can be defined on the toplevel or (2) they can appear inside type classes (in which case they are known as associated types). The former is the more general variant, as it lacks the requirement for the type-indices to coincide with the class parameters. However, the latter can lead to more clearly structured code and compiler warnings if some type instances were - possibly accidentally - omitted. In the following, we always discuss the general toplevel form first and then cover the additional constraints placed on associated types.

4.1 Family declarations

Indexed data families are introduced by a signature, such as

data family GMap k ::*->*

The special

family

distinguishes family from standard data declarations. The result kind annotation is optional and, as usual, defaults to

*

if omitted. An example is

data family Array e

Named arguments can also be given explicit kind signatures if needed. Just as with GADT declarations named arguments are entirely optional, so that we can declare

Array

alternatively with

data family Array ::*->*

4.1.1 Associated family declarations

When a data family is declared as part of a type class, we drop the

family

keyword. The

GMap

declaration takes the following form

class GMapKey k wheredata GMap k ::*->*...

In contrast to toplevel declarations, named arguments must be used for all type parameters that are to be used as type-indices. Moreover, the argument names must be class parameters. Each class parameter may only be used at most once per associated type, but some may be omitted and they may be in an order other than in the class head. In other words: the named type parameters of the data declaration must be a permutation of a subset of the class variables.

4.2 Instance declarations

Instance declarations of data and newtype families are very similar to standard data and newtype declarations. The only two differences are that the keyword

data

or

newtype

is followed by

instance

and that some or all of the type arguments can be non-variable types, but may not contain forall types or type synonym families. However, data families are generally allowed in type parameters, and type synonyms are allowed as long as they are fully applied and expand to a type that is itself admissible - exactly as this is required for occurrences of type synonyms in class instance parameters. For example, the

Either

instance for

GMap

is

datainstance GMap (Either a b) v = GMapEither (GMap a v)(GMap b v)

In this example, the declaration has only one variant. In general, it can be any number.

Data and newtype instance declarations are only legit when an appropriate family declaration is in scope - just like class instances require the class declaration to be visible. Moreover, each instance declaration has to conform to the kind determined by its family declaration. This implies that the number of parameters of an instance declaration matches the arity determined by the kind of the family. Although all data families are declared with the

data

keyword, instances can be either

data

or

newtype

s, or a mix of both.

Even if type families are defined as toplevel declarations, functions that perform different computations for different family instances still need to be defined as methods of type classes. In particular, the following is not possible:

Given the functionality provided by GADTs (Generalised Algebraic Data Types), it might seem as if a definition, such as the above, should be feasible. However, type families - in contrast to GADTs - are open; i.e., new instances can always be added, possibly in other modules. Supporting pattern matching across different data instances would require a form of extensible case construct.

4.2.1 Associated type instances

When an associated family instance is declared within a type class instance, we drop the

The most important point about associated family instances is that the type indices corresponding to class parameters must be identical to the type given in the instance head; here this is the first argument of

GMap

, namely

Either a b

, which coincides with the only class parameter. Any parameters to the family constructor that do not correspond to class parameters, need to be variables in every instance; here this is the variable

v

.
Instances for an associated family can only appear as part of instance declarations of the class in which the family was declared - just as with the equations of the methods of a class. Also in correspondence to how methods are handled, declarations of associated types can be omitted in class instances. If an associated family instance is omitted, the corresponding instance type is not inhabited; i.e., only diverging expressions, such as

undefined

, can assume the type.

4.2.2 Scoping of class parameters

In the case of multi-parameter type classes, the visibility of class parameters in the right-hand side of associated family instances depends solely on the parameters of the data family. As an example, consider the simple class declaration

class C a b wheredata T a

Only one of the two class parameters is a parameter to the data family. Hence, the following instance declaration is invalid:

Here, the right-hand side of the data instance mentions the type variable

d

that does not occur in its left-hand side. We cannot admit such data instances as they would compromise type safety.

4.2.3 Type class instances of family instances

Type class instances of instances of data families can be defined as usual, and in particular data instance declarations can have

deriving

clauses. For example, we can write

data GMap () v = GMapUnit (Maybe v)derivingShow

which implicitly defines an instance of the form

instanceShow v =>Show(GMap () v)where...

Note that class instances are always for particular instances of a data family and never for an entire family as a whole. This is for essentially the same reasons that we cannot define a toplevel function that performs pattern matching on the data constructors of different instances of a single type family. It would require a form of extensible case construct.

4.2.4 Overlap

The instance declarations of a data family used in a single program may not overlap at all, independent of whether they are associated or not. In contrast to type class instances, this is not only a matter of consistency, but one of type safety.

4.3 Import and export

The association of data constructors with type families is more dynamic than that is the case with standard data and newtype declarations. In the standard case, the notation

T(..)

in an import or export list denotes the type constructor and all the data constructors introduced in its declaration. However, a family declaration never introduces any data constructors; instead, data constructors are introduced by family instances. As a result, which data constructors are associated with a type family depends on the currently visible instance declarations for that family. Consequently, an import or export item of the form

T(..)

denotes the family constructor and all currently visible data constructors - in the case of an export item, these may be either imported or defined in the current module. The treatment of import and export items that explicitly list data constructors, such as

GMap(GMapEither)

, is analogous.

4.3.1 Associated families

As expected, an import or export item of the form

C(..)

denotes all of the class' methods and associated types. However, when associated types are explicitly listed as subitems of a class, we need some new syntax, as uppercase identifiers as subitems are usually data constructors, not type constructors. To clarify that we denote types here, each associated type name needs to be prefixed by the keyword

type

. So for example, when explicitly listing the components of the

GMapKey

class, we write

GMapKey(type GMap, empty,lookup, insert)

.

4.3.2 Examples

Assuming our running

GMapKey

class example, let us look at some export lists and their meaning:

module GMap (GMapKey)where...

: Exports just the class name.

module GMap (GMapKey(..))where...

: Exports the class, the associated type

GMap

and the member functions

empty

,

lookup

, and

insert

. None of the data constructors is exported.

module GMap (GMapKey(..), GMap(..))where...

: As before, but also exports all the data constructors

GMapInt

,

GMapChar

,

GMapUnit

,

GMapPair

, and

GMapEither

.

module GMap (GMapKey(empty,lookup, insert), GMap(..))where...

: As before.

module GMap (GMapKey, empty,lookup, insert, GMap(..))where...

: As before.

Finally, you can write

GMapKey(type GMap)

to denote both the class

GMapKey

as well as its associated type

GMap

. However, you cannot write

GMapKey(type GMap(..))

— i.e., sub-component specifications cannot be nested. To specify

GMap

's data constructors, you have to list it separately.

4.3.3 Instances

Family instances are implicitly exported, just like class instances. However, this applies only to the heads of instances, not to the data constructors an instance defines.

5 An associated type synonym example

Type synonym families are an alternative to functional dependencies, which makes functional dependency examples well suited to introduce type synonym families. In fact, type families are a more functional way to express the same as functional dependencies (despite the name!), as they replace the relational notation of functional dependencies by an expression-oriented notation; i.e., functions on types are really represented by functions and not relations.

6 Detailed definition of type synonym families

Type families appear in two flavours: (1) they can be defined on the toplevel or (2) they can appear inside type classes (in which case they are known as associated type synonyms). The former is the more general variant, as it lacks the requirement for the type-indices to coincide with the class parameters. However, the latter can lead to more clearly structured code and compiler warnings if some type instances were - possibly accidentally - omitted. In the following, we always discuss the general toplevel form first and then cover the additional constraints placed on associated types.

6.1 Family declarations

Indexed type families are introduced by a signature, such as

type family Elem c ::*

The special

family

distinguishes family from standard type declarations. The result kind annotation is optional and, as usual, defaults to

*

if omitted. An example is

type family Elem c

Parameters can also be given explicit kind signatures if needed. We call the number of parameters in a type family declaration, the family's arity, and all applications of a type family must be fully saturated w.r.t. to that arity. This requirement is unlike ordinary type synonyms and it implies that the kind of a type family is not sufficient to determine a family's arity, and hence in general, also insufficient to determine whether a type family application is well formed. As an example, consider the following declaration:

A top-level type family can be declared as open or closed. (Associated type
families are always open.) A closed type family has all of its equations
defined in one place and cannot be extended, whereas an open family can have
instances spread across modules. The advantage of a closed family is that
its equations are tried in order, similar to a term-level function definition:

6.2 Type instance declarations

Instance declarations of open type families are very similar to standard type synonym declarations. The only two differences are that the keyword

type

is followed by

instance

and that some or all of the type arguments can be non-variable types, but may not contain forall types or type synonym families. However, data families are generally allowed, and type synonyms are allowed as long as they are fully applied and expand to a type that is admissible - these are the exact same requirements as for data instances. For example, the

[e]

instance for

Elem

is

typeinstance Elem [e]= e

A type family instance declaration must satisfy the following rules:

An appropriate family declaration is in scope - just like class instances require the class declaration to be visible.

The instance declaration conforms to the kind determined by its family declaration

The number of type parameters in an instance declaration matches the number of type parameters in the family declaration.

The right-hand side of a type instance must be a monotype (i.e., it may not include foralls) and after the expansion of all saturated vanilla type synonyms, no synonyms, except family synonyms may remain.

Here are some examples of admissible and illegal type instances and closed families:

6.2.1 Closed family simplification

Currently in recent versions of ghc 7.7 and planed to be included in 7.8.1.

When dealing with closed families, simplifying the type is harder than just finding a left-hand side that matches and replacing that with a right-hand side. GHC will select an equation to use in a given type family application (the "target") if and only if the following 2 conditions hold:

There is a substitution from the variables in the equation's LHS that makes the left-hand side of the branch coincide with the target.

For each previous equation in the family: either the LHS of that equation is apart from the type family application, or the equation is compatible with the chosen equation.

Now, we define apart and compatible:

Two types are apart when one cannot simplify to the other, even after arbitrary type-family simplifications

Two equations are compatible if, either, their LHSs are apart or their LHSs unify and their RHSs are the same under the substitution induced by the unification.

, all pairs of equations are compatible except the second and third. The first two are compatible because their LHSs are apart. The first and third are compatible because the unifying substitution leads the RHSs to be the same. But, the second and third are not compatible because neither of these conditions holds. As a result, GHC will not use the third equation to simplify a target unless that target is apart from

Bool

.
In

And

, every pair of equations is compatible, meaning GHC never has to make the extra apartness check during simplification.

Why do all of this? It's a matter of type safety. Consider this example:

type family J a b
typeinstancewhere
J a a =Int
J a b =Bool

Say GHC selected the second branch just because the first doesn't apply at the moment, because two type variables are distinct. The problem is that those variables might later be instantiated at the same value, and then the first branch would have applied. You can convince this sort of inconsistency to produce

unsafeCoerce

.

It gets worse. GHC has no internal notion of inequality, so it can't use previous, failed term-level GADT pattern matches to refine its type assumptions. For example:

The last line will fail to typecheck, because GHC doesn't know that the type variable

a

can't be

Int

here, even though it's obvious. The only general way to fix this is to have inequality evidence introduced into GHC, and that's a big deal and we don't know if we have the motivation for such a change yet.

6.2.2 Associated type instances

When an associated family instance is declared within a type class instance, we drop the

instance

keyword in the family instance. So, the

[e]

instance for

Elem

becomes:

instance(Eq(Elem [e]))=> Collects ([e])wheretype Elem [e]= e
...

The most important point about associated family instances is that the type indexes corresponding to class parameters must be identical to the type given in the instance head; here this is

[e]

, which coincides with the only class parameter.
Instances for an associated family can only appear as part of instance declarations of the class in which the family was declared - just as with the equations of the methods of a class. Also in correspondence to how methods are handled, declarations of associated types can be omitted in class instances. If an associated family instance is omitted, the corresponding instance type is not inhabited; i.e., only diverging expressions, such as

undefined

, can assume the type.

6.2.3 Overlap

The instance declarations of an open type family used in a single program must be compatible, in the form defined above. This condition is independent of whether the type family is associated or not, and it is not only a matter of consistency, but one of type safety.

Here are two examples to illustrate the condition under which overlap is permitted.

6.2.4 Decidability

In order to guarantee that type inference in the presence of type families is decidable, we need to place a number of additional restrictions on the formation of type instance declarations (c.f., Definition 5 (Relaxed Conditions) of Type Checking with Open Type Functions). Instance declarations have the general form

typeinstance F t1 .. tn = t

where we require that for every type family application

(G s1 .. sm)

in

t

,

s1 .. sm

do not contain any type family constructors,

the total number of symbols (data type constructors and type variables) in

s1 .. sm

is strictly smaller than in

t1 .. tn

, and

for every type variable

a

,

a

occurs in

s1 .. sm

at most as often as in

t1 .. tn

.

These restrictions are easily verified and ensure termination of type inference. However, they are not sufficient to guarantee completeness of type inference in the presence of, so called, loopy equalities, such as

a ~[F a]

, where a recursive occurrence of a type variable is underneath a family application and data constructor application - see the above mentioned paper for details.

If the option -XUndecidableInstances is passed to the compiler, the above restrictions are not enforced and it is on the programmer to ensure termination of the normalisation of type families during type inference.

6.3 Equality constraints

Type context can include equality constraints of the form

t1 ~ t2

, which denote that the types

t1

and

t2

need to be the same. In the presence of type families, whether two types are equal cannot generally be decided locally. Hence, the contexts of function signatures may include equality constraints, as in the following example:

of an equality constraint may be arbitrary monotypes; i.e., they may not contain any quantifiers, independent of whether higher-rank types are otherwise enabled.

Equality constraints can also appear in class and instance contexts. The former enable a simple translation of programs using functional dependencies into programs using family synonyms instead. The general idea is to rewrite a class declaration of the form

class C a b | a -> b

to

class(F a ~ b)=> C a b wheretype F a

That is, we represent every functional dependency (FD)

a1 .. an -> b

by an FD type family

F a1 .. an

and a superclass context equality

F a1 .. an ~ b

, essentially giving a name to the functional dependency. In class instances, we define the type instances of FD families in accordance with the class head. Method signatures are not affected by that process.

7 Frequently asked questions

7.1 Comparing type families and functional dependencies

Functional dependencies cover some of the same territory as type families. How do the two compare?
There are some articles about this question:

In type-checking g's right hand side GHC discovers (by instantiating f's type with a fresh type variable) that it has type F a1 -> F a1 for some as-yet-unknown type a1. Now it tries to make the inferred type match g's type signature. Well, you say, just make a1 equal to Int and you are done. True, but what if there were these instances

typeinstance F Int=Booltypeinstance F Char=Bool

Then making a1 equal to Char would also make the two types equal. Because there is (potentially) more than one choice, the program is rejected.

However (and confusingly) if you omit the type signature on g altogether, thus

f :: F a -> F a
f =undefined
g x = f x

GHC will happily infer the type g :: F a -> F a. But you can't write that type signature or, indeed, the more specific one above. (Arguably this behaviour, where GHC infers a type it can't check, is very confusing. I suppose we could make GHC reject both programs, with and without type signatures.)

What is the problem? The nub of the issue is this: knowing that F t1=F t2 does not imply that t1 = t2.
The difficulty is that the type function F need not be injective; it can map two distinct types to the same type. For an injective type constructor like Maybe, if we know that Maybe t1 = Maybe t2, then we know that t1 = t2. But not so for non-injective type functions.

The problem starts with f. Its type is ambiguous; even if I know the argument and result types for f, I cannot use that to find the type at which a should be instantiated. (So arguably, f should be rejected as having an ambiguous type, and probably will be in future.) The situation is well known in type classes:

bad ::(Read a,Show a)=>String->String
bad x =show(read x)

At a call of bad one cannot tell at what type a should be instantiated.

The only solution is to avoid ambiguous types. In the type signature of a function,

Ensure that every type variable occurs in the part after the "=>"

Ensure that every type variable appears at least once outside a type function call.

Alternatively, you can use data families, which create new types and are therefore injective. The following code works: